Ch 3.1: Second Order Linear Homogeneous Equations with Constant Coefficients

1. Ch 3.1: Second Order Linear Homogeneous Equations with Constant Coefficients • A second order ordinary differential equation has the general form where f is some given function. • This equation is said to be linear if f is linear in y and y': Otherwise the equation is said to be nonlinear. • A second order linear equation often appears as • If G(t) = 0 for all t, then the equation is called homogeneous. Otherwise the equation is nonhomogeneous.

2. Homogeneous Equations, Initial Values • In Sections 3.6 and 3.7, we will see that once a solution to a homogeneous equation is found, then it is possible to solve the corresponding nonhomogeneous equation, or at least express the solution in terms of an integral. • The focus of this chapter is thus on homogeneous equations; and in particular, those with constant coefficients: We will examine the variable coefficient case in Chapter 5. • Initial conditions typically take the form • Thus solution passes through (t0, y0), and slope of solution at (t0, y0) is equal to y0'.

3. Example 1: Infinitely Many Solutions (1 of 3) • Consider the second order linear differential equation • Two solutions of this equation are • Other solutions include • Based on these observations, we see that there are infinitely many solutions of the form • It will be shown in Section 3.2 that all solutions of the differential equation above can be expressed in this form.

4. Example 1: Initial Conditions (2 of 3) • Now consider the following initial value problem for our equation: • We have found a general solution of the form • Using the initial equations, • Thus

5. Example 1: Solution Graphs (3 of 3) • Our initial value problem and solution are • Graphs of this solution are given below. The graph on the right suggests that both initial conditions are satisfied.

6. Characteristic Equation • To solve the 2nd order equation with constant coefficients, we begin by assuming a solution of the form y = ert. • Substituting this into the differential equation, we obtain • Simplifying, and hence • This last equation is called the characteristic equation of the differential equation. • We then solve for r by factoring or using quadratic formula.

7. General Solution • Using the quadratic formula on the characteristic equation we obtain two solutions, r1 and r2. • There are three possible results: • The roots r1, r2 are real and r1r2. • The roots r1, r2 are real and r1 = r2. • The roots r1, r2 are complex. • In this section, we will assume r1, r2 are real and r1r2. • In this case, the general solution has the form

8. Initial Conditions • For the initial value problem we use the general solution together with the initial conditions to find c1 and c2. That is, • Since we are assuming r1r2, it follows that a solution of the form y = ert to the above initial value problem will always exist, for any set of initial conditions.

9. Example 2 • Consider the initial value problem • Assuming exponential soln leads to characteristic equation: • Factoring yields two solutions, r1 = -4 and r2 = 3 • The general solution has the form • Using the initial conditions: • Thus

10. Example 3 • Consider the initial value problem • Then • Factoring yields two solutions, r1 = 0 and r2 = -3/2 • The general solution has the form • Using the initial conditions: • Thus

11. Example 4: Initial Value Problem (1 of 2) • Consider the initial value problem • Then • Factoring yields two solutions, r1 = -2 and r2 = -3 • The general solution has the form • Using initial conditions: • Thus

12. Example 4: Find Maximum Value (2 of 2) • Find the maximum value attained by the solution.